zbMATH — the first resource for mathematics

Design of the control set in the framework of variational data assimilation. (English) Zbl 1375.93072
Summary: Solving data assimilation problems under uncertainty in basic model parameters and in source terms may require a careful design of the control set. The task is to avoid such combinations of the control variables which may either lead to ill-posedness of the control problem formulation or compromise the robustness of the solution procedure. We suggest a method for quantifying the performance of a control set which is formed as a subset of the full set of uncertainty-bearing model inputs. Based on this quantity one can decide if the chosen ‘safe’ control set is sufficient in terms of the prediction accuracy. Technically, the method presents a certain generalization of the ‘variational’ uncertainty quantification method for observed systems. It is implemented as a matrix-free method, thus allowing high-dimensional applications. Moreover, if the automatic differentiation is utilized for computing the tangent linear and adjoint mappings, then it could be applied to any multi-input ‘black-box’ system. As application example we consider the full Saint-Venant hydraulic network model SIC\(^{2}\), which describes the flow dynamics in river and canal networks. The developed methodology seem useful in the context of the future SWOT satellite mission, which will provide observations of river systems the properties of which are known with quite a limited precision.

93C41 Control/observation systems with incomplete information
65D25 Numerical differentiation
Full Text: DOI
[1] Abdolghafoorian, A.; Farhadi, L., Uncertainty quantification in land surface hydrologic modeling: toward an integrated variational data assimilation framework, IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., 9, 6, 2628-2637, (2016)
[2] Akella, S.; Navon, I. M., Different approaches to model error formulation in 4D-var: a study with high resolution advection schemes, Tellus, 61A, 112-128, (2009)
[3] Alifanov, O. M.; Artyukhin, E. A.; Rumyantsev, S. V., Extreme methods for solving ill-posed problems with applications to inverse heat transfer problems, (1996), Begel House Publishers · Zbl 1006.35001
[4] Auvinen, H.; Bardsley, J. M.; Haario, H.; Kauranne, T., The variational Kalman filter and an efficient implementation using limited memory BFGS, Int. J. Numer. Methods Fluids, 64, 3, 314-335, (2010) · Zbl 1197.65213
[5] Burstedde, C.; Ghattas, O., Algorithmic strategies for full waveform inversion: 1d experiments, Geophysics, 74, WCC37-WCC46, (2009)
[6] Courtier, P.; Thepaut, J.-N.; Hollingsworth, A., A strategy for operational implementation of 4d-var, using an incremental approach, Q. J. R. Meteorol. Soc., 120, 1367-1387, (1994)
[7] Dashti, M.; Law, K. J.H.; Stuart, A. M.; Voss, J., Map estimators and posterior consistency in Bayesian nonparametric inverse problems, Inverse Probl., 29, (2013) · Zbl 1281.62089
[8] Davies, E. G.R.; Simonovic, S. P., Global water resources modeling with an integrated model of the social-economic-environmental system, Adv. Water Resour., 34, 6, 684-700, (2011)
[9] Fernandes, J. N.; Cardoso, A. H., Flow structure in a compound channel with smooth and rough floodplains, EWRA Eur. Water Publ., 38, 3-12, (2012)
[10] Gejadze, I.; Le Dimet, F.-X.; Shutyaev, V., On optimal solution error covariances in variational data assimilation problems, J. Comput. Phys., 229, 6, 2159-2178, (2010) · Zbl 1185.65106
[11] Gejadze, I.; Malaterre, P.-O., Discharge estimation under uncertainty using variational methods with application to the full Saint-Venant hydraulic network model, Int. J. Numer. Methods Fluids, 34, 127-147, (2016)
[12] Hascoët, L.; Pascual, V., Tapenade 2.1 User’s guide, (2004), INRIA Technical Report, 0300
[13] Hoang, H.; Baraille, R., Stochastic simultaneous perturbation as powerful method for state and parameter estimation in high dimensional systems, (Advances in Mathematical Research, vol. 20, (2015), Nova Science Publishers), 117-148
[14] Isaac, T.; Petra, N.; Stadler, G.; Ghattas, O., Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the antarctic ice sheet, J. Comput. Phys., 296, 348-368, (2015) · Zbl 1352.86017
[15] Jazwinski, A. H., Stochastic processes and filtering theory, (1970), Academic Press · Zbl 0203.50101
[16] Kalmikov, A. G.; Heimbach, P., A hessian-based method for uncertainty quantification in global Ocean statez estimation, SIAM J. Sci. Comput., 36, 5, S267-S295, (2014) · Zbl 1311.35321
[17] Le Dimet, F.-X.; Talagrand, O., Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects, Tellus A, 38A, 2, 97-110, (1986)
[18] Lieberman, C.; Wilcox, K., Goal-oriented inference: approach, linear theory, and application to advection diffusion, SIAM J. Sci. Comput., 34, 4, A1880-A1904, (2012) · Zbl 1250.62058
[19] Lions, J.-L., Contrôle optimal des systèmes gouvernés par des équations aux Dérivées partielles, (1968), Dunod Paris · Zbl 0179.41801
[20] Liu, C.; Xiao, Q.; Wang, B., An ensemble-based four-dimensional variational data assimilation scheme. part I: technical formulation and preliminary test, Mon. Weather Rev., 136, 3363-3373, (2008)
[21] Honnorat, M.; Monnier, J.; Le Dimet, F.-X., Lagrangian data assimilation for river hydraulics simulations, Comput. Vis. Sci., 12, 5, 235-246, (2009) · Zbl 1426.86005
[22] Malaterre, P.-O.; Baume, J.-P.; Dorchies, D., Simulation and integration of control for canals software (\(s i c^2\)), for the design and verification of manual or automatic controllers for irrigation canals, (USCID Conference on Planning, Operation and Automation of Irrigation Delivery Systems, Phoenix, Arizona, December 2-5, (2014)), 377-382
[23] Malaterre, P.-O.; Khammash, M., \(\ell_1\) controller design for a high-order 5-pool irrigation canal system, ASME J. Dyn. Syst. Meas. Control, 125, 639-645, (2003)
[24] Marchuk, G. I.; Agoshkov, V. I.; Shutyaev, V. P., Adjoint equations and perturbation algorithms in nonlinear problems, (1996), CRC Press Inc. New York
[25] Martin, N.; Monnier, J., Adjoint accuracy for the full-Stokes ice flow model: limits to the transmission of basal friction variability to the surface, Cryosphere, 8, 721-741, (2014)
[26] Moireau, P.; Chapelle, D., Reduced-order unscented Kalman filtering with application to parameter identification in large-dimensional systems, ESAIM Control Optim. Calc. Var., 17, 2, 380-405, (2011) · Zbl 1243.93114
[27] Navon, I. M., Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography, Dyn. Atmos. Ocean., 27, 1-4, 55-79, (1998)
[28] Novak, P.; Guinot, V.; Jeffrey, A.; Reeve, D. E., Hydraulic modelling - an introduction: principles, methods and applications, (2010), CRC Press
[29] Sarma, P.; Durlofsky, L. J.; Aziz, H.; Chen, W. H., Efficient real-time reservoir management using adjoint-based optimal control and model updating, Comput. Geosci., 10, 3-36, (2005) · Zbl 1161.86303
[30] Sart, C.; Baume, J.-P.; Malaterre, P.-O.; Guinot, V., Adaptation of Preissmann’s scheme for transcritical open channel flows, J. Hydraul. Res., 48, 4, 428-440, (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.